464 Test Bank Questions and Answers TEST BANK Questions for Chapter What is the negation of the propositions in 1–3? Abby has more than 300 friends on facebook A messaging package for a cell phone costs less than $20 per month 4.5 + 2.5 = In questions 4–8, determine whether the proposition is TRUE or FALSE + = if and only if + = If it is raining, then it is raining If < 0, then = If + = 3, then = − If + = or + = 3, then + = and + = Write the truth table for the proposition ¬(r → ¬q) ∨ (p ∧ ¬r) 10 (a) Find a proposition with the truth table at the right (b) Find a proposition using only p, q, ¬, and the connective ∨ that has this truth table p T T F F q T F T F ? F F T F 11 Find a proposition with three variables p, q, and r that is true when p and r are true and q is false, and false otherwise 12 Find a proposition with three variables p, q, and r that is true when at most one of the three variables is true, and false otherwise 13 Find a proposition with three variables p, q, and r that is never true 14 Find a proposition using only p, q, ¬, and the connective ∨ with the truth table at the right 15 Determine whether p → (q → r) and p → (q ∧ r) are equivalent 16 Determine whether p → (q → r) is equivalent to (p → q) → r 17 Determine whether (p → q) ∧ (¬p → q) ≡ q p T T F F q T F T F ? F T T F Test Bank Questions and Answers 465 18 Write a proposition equivalent to p ∨ ¬q that uses only p, q, ¬, and the connective ∧ 19 Write a proposition equivalent to ¬p ∧ ¬q using only p, q, ¬, and the connective ∨ 20 Prove that the proposition “if it is not hot, then it is hot” is equivalent to “it is hot” 21 Write a proposition equivalent to p → q using only p, q, ¬, and the connective ∨ 22 Write a proposition equivalent to p → q using only p, q, ¬, and the connective ∧ 23 Prove that p → q and its converse are not logically equivalent 24 Prove that ¬p → ¬q and its inverse are not logically equivalent 25 Determine whether the following two propositions are logically equivalent: p ∨ (q ∧ r), (p ∧ q) ∨ (p ∧ r) 26 Determine whether the following two propositions are logically equivalent: p → (¬q ∧ r), ¬p ∨ ¬(r → q) 27 Prove that (q ∧ (p → ¬q)) → ¬p is a tautology using propositional equivalence and the laws of logic 28 Determine whether this proposition is a tautology: ((p → q) ∧ ¬p) → ¬q 29 Determine whether this proposition is a tautology: ((p → ¬q) ∧ q) → ¬p In 30–36, write the statement in the form “If , then ” 30 x is even only if y is odd 31 A implies B 32 It is hot whenever it is sunny 33 To get a good grade it is necessary that you study 34 Studying is sufficient for passing 35 The team wins if the quarterback can pass 36 You need to be registered in order to check out library books 37 Write the contrapositive, converse, and inverse of the following: If you try hard, then you will win 38 Write the contrapositive, converse, and inverse of the following: You sleep late if it is Saturday In 39–41 write the negation of the statement (Don’t write “It is not true that ”) 39 It is Thursday and it is cold 40 I will go to the play or read a book, but not both 41 If it is rainy, then we go to the movies 42 Explain why the negation of “Al and Bill are absent” is not “Al and Bill are present” 43 Using c for “it is cold” and d for “it is dry”, write “It is neither cold nor dry” in symbols 44 Using c for “it is cold” and r for “it is rainy”, write “It is rainy if it is not cold” in symbols 45 Using c for “it is cold” and w for “it is windy”, write “To be windy it is necessary that it be cold” in symbols 46 Using c for “it is cold”, r for “it is rainy”, and w for “it is windy”, write “It is rainy only if it is windy and cold” in symbols 466 Test Bank Questions and Answers 47 Translate the given statement into propositional logic using the propositions provided: On certain highways in the Washington, DC metro area you are allowed to travel on high occupancy lanes during rush hour only if there are at least three passengers in the vehicle Express your answer in terms of r:“You are traveling during rush hour.” t:“You are riding in a car with at least three passengers.” and h:“You can travel on a high occupancy lane.” 48 A set of propositions is consistent if there is an assignment of truth values to each of the variables in the propositions that makes each proposition true Is the following set of propositions consistent? The system is in multiuser state if and only if it is operating normally If the system is operating normally, the kernel is functioning The kernel is not functioning or the system is in interrupt mode If the system is not in multiuser state, then it is in interrupt mode The system is in interrupt mode 49 On the island of knights and knaves you encounter two people, A and B Person A says “B is a knave.” Person B says “We are both knights.” Determine whether each person is a knight or a knave 50 On the island of knights and knaves you encounter two people, A and B Person A says “B is a knave.” Person B says “At least one of us is a knight.” Determine whether each person is a knight or a knave Exercises 51–53 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies who can either tell the truth or lie You encounter three people, A, B, and C You know one of the three people is a knight, one is a knave, and one is a spy Each of the three people knows the type of person each of the other two is For each of these situations, if possible, determine whether there is a unique solution, list all possible solutions or state that there are no solutions 51 A says “I am not a knight,” B says “I am not a spy,” and C says “I am not a knave.” 52 A says “I am a spy,” B says “I am a spy” and C says “B is a spy.” 53 A says “I am a knight,” B says “I am a knave,” and C says “I am not a knave.” Find the output of the combinatorial circuits in 54–55 54 p q r 55 p q q r Construct a combinatorial circuit using inverters, OR gates, and AND gates, that produces the outputs in 56–57 from input bits p, q and r 56 (¬p ∧ ¬q) ∨ (p ∧ ¬r) 57 ((p ∨ ¬q) ∧ r) ∧ ((¬p ∧ ¬q) ∨ r) Determine whether the compound propositions in 58–59 are satisfiable 58 (¬p ∨ ¬q) ∧ (p → q) 59 (p → q) ∧ (q → ¬p) ∧ (p ∨ q) In 60–62 suppose that Q(x) is “x + = 2x”, where x is a real number Find the truth value of the statement Test Bank Questions and Answers 467 60 Q(2) 61 ∀x Q(x) 62 ∃x Q(x) In 63–70 P (x, y) means “x+2y = xy”, where x and y are integers Determine the truth value of the statement 63 P (1, −1) 64 P (0, 0) 65 ∃y P (3, y) 66 ∀x∃y P (x, y) 67 ∃x∀y P (x, y) 68 ∀y∃x P (x, y) 69 ∃y∀x P (x, y) 70 ¬∀x∃y ¬P (x, y) In 71–72 P (x, y) means “x and y are real numbers such that x + 2y = 5” Determine whether the statement is true 71 ∀x∃y P (x, y) 72 ∃x∀y P (x, y) In 73–75 P (m, n) means “m ≤ n”, where the universe of discourse for m and n is the set of nonnegative integers What is the truth value of the statement? 73 ∀n P (0, n) 74 ∃n∀m P (m, n) 75 ∀m∃n P (m, n) In questions 76–81 suppose P (x, y) is a predicate and the universe for the variables x and y is {1, 2, 3} Suppose P (1, 3), P (2, 1), P (2, 2), P (2, 3), P (3, 1), P (3, 2) are true, and P (x, y) is false otherwise Determine whether the following statements are true 76 ∀x∃yP (x, y) 77 ∃x∀yP (x, y) 78 ¬∃x∃y (P (x, y) ∧ ¬P (y, x)) 79 ∀y∃x (P (x, y) → P (y, x)) 80 ∀x∀y (x = y → (P (x, y) ∨ P (y, x)) 81 ∀y∃x (x ≤ y ∧ P (x, y)) In 82–85 suppose the variable x represents students and y represents courses, and: U (y): y is an upper-level course M (y): y is a math course F (x): x is a freshman B(x): x is a full-time student T (x, y): student x is taking course y Write the statement using these predicates and any needed quantifiers 82 Eric is taking MTH 281 468 Test Bank Questions and Answers 83 All students are freshmen 84 Every freshman is a full-time student 85 No math course is upper-level In 86–88 suppose the variable x represents students and y represents courses, and: U (y): y is an upper-level course M (y): y is a math course F (x): x is a freshman A(x): x is a part-time student T (x, y): student x is taking course y Write the statement using these predicates and any needed quantifiers 86 Every student is taking at least one course 87 There is a part-time student who is not taking any math course 88 Every part-time freshman is taking some upper-level course In 89–91 suppose the variable x represents students and y represents courses, and: F (x): x is a freshman A(x): x is a part-time student T (x, y): x is taking y Write the statement in good English without using variables in your answers 89 F (Mikko) 90 ¬∃y T (Joe, y) 91 ∃x (A(x) ∧ ¬F (x)) In 92–94 suppose the variable x represents students and y represents courses, and: M (y): y is a math course F (x): x is a freshman B(x): x is a full-time student T (x, y): x is taking y Write the statement in good English without using variables in your answers 92 ∀x∃y T (x, y) 93 ∃x∀y T (x, y) 94 ∀x∃y [(B(x) ∧ F (x)) → (M (y) ∧ T (x, y))] In 95–97 suppose the variables x and y represent real numbers, and L(x, y) : x < y G(x) : x > P (x) : x is a prime number Write the statement in good English without using any variables in your answer 95 L(7, 3) 96 ∀x∃y L(x, y) 97 ∀x∃y [G(x) → (P (y) ∧ L(x, y))] In 98–100 suppose the variables x and y represent real numbers, and L(x, y) : x < y Q(x, y) : x = y E(x) : x is even Write the statement using these predicates and any needed quantifiers 98 Every integer is even 99 If x < y, then x is not equal to y 100 There is no largest real number In 101–102 suppose the variables x and y represent real numbers, and I(x) : x is an integer 469 Test Bank Questions and Answers E(x) : x is even G(x) : x > I(x) : x is an integer Write the statement using these predicates and any needed quantifiers 101 Some real numbers are not positive 102 No even integers are odd In 103–105 suppose the variable x represents people, and F (x): x is friendly T (x): x is tall A(x): x is angry Write the statement using these predicates and any needed quantifiers 103 Some people are not angry 104 All tall people are friendly 105 No friendly people are angry In 106–107 suppose the variable x represents people, and F (x): x is friendly T (x): x is tall A(x): x is angry Write the statement using these predicates and any needed quantifiers 106 Some tall angry people are friendly 107 If a person is friendly, then that person is not angry In 108–110 suppose the variable x represents people, and F (x): x is friendly T (x): x is tall A(x): x is angry Write the statement in good English Do not use variables in your answer 108 A(Bill) 109 ¬∃x (A(x) ∧ T (x)) 110 ¬∀x (F (x) → A(x)) In 111–113 suppose the variable x represents students and the variable y represents courses, and A(y): y is an advanced course S(x): x is a sophomore F (x): x is a freshman T (x, y): x is taking y Write the statement using these predicates and any needed quantifiers 111 There is a course that every freshman is taking 112 No freshman is a sophomore 113 Some freshman is taking an advanced course In 114–115 suppose the variable x represents students and the variable y represents courses, and A(y): y is an advanced course F (x): x is a freshman T (x, y): x is taking y P (x, y): x passed y Write the statement using the above predicates and any needed quantifiers 114 No one is taking every advanced course 115 Every freshman passed calculus In 116–118 suppose the variable x represents students and the variable y represents courses, and T (x, y): x is taking y P (x, y): x passed y Write the statement in good English Do not use variables in your answers 116 ¬P (Wisteria, MAT 100) 117 ∃y∀x T (x, y) 470 Test Bank Questions and Answers 118 ∀x∃y T (x, y) In 119–123 assume that the universe for x is all people and the universe for y is the set of all movies Write the English statement using the following predicates and any needed quantifiers: S(x, y): x saw y L(x, y): x liked y A(y): y won an award C(y): y is a comedy 119 No comedy won an award 120 Lois saw Casablanca, but didn’t like it 121 Some people have seen every comedy 122 No one liked every movie he has seen 123 Ben has never seen a movie that won an award In 124–126 assume that the universe for x is all people and the universe for y is the set of all movies Write the statement in good English, using the predicates S(x, y): x saw y L(x, y): x liked y Do not use variables in your answer 124 ∃y ¬S(Margaret, y) 125 ∃y∀x L(x, y) 126 ∀x∃y L(x, y) In 127–136 suppose the variable x represents students, y represents courses, and T (x, y) means “x is taking y” Match the English statement with all its equivalent symbolic statements in this list: ∃x∀y T (x, y) ¬∃x∃y T (x, y) ∃y∀x ¬T (x, y) 10 ¬∀x∃y ¬T (x, y) ∃y∀x T (x, y) ∃x∀y ¬T (x, y) ¬∀x∃y T (x, y) 11 ¬∀x¬∀y ¬T (x, y) 127 Every course is being taken by at least one student ∀x∃y T (x, y) ∀y∃x T (x, y) ¬∃y∀x T (x, y) 12 ∀x∃y ¬T (x, y) 128 Some student is taking every course 129 No student is taking all courses 130 There is a course that all students are taking 131 Every student is taking at least one course 132 There is a course that no students are taking 133 Some students are taking no courses 134 No course is being taken by all students 135 Some courses are being taken by no students 136 No student is taking any course In 137–147 suppose the variable x represents students, F (x) means “x is a freshman’,’ and M (x) means “x is a math major” Match the statement in symbols with one of the English statements in this list: Some freshmen are math majors Every math major is a freshman No math major is a freshman Test Bank Questions and Answers 471 137 ∀x (M (x) → ¬F (x)) 138 ¬∃x (M (x) ∧ ¬F (x)) 139 ∀x (F (x) → ¬M (x)) 140 ∀x (M (x) → F (x)) 141 ∃x (F (x) ∧ M (x)) 142 ¬∀x (¬F (x) ∨ ¬M (x)) 143 ∀x (¬(M (x) ∧ ¬F (x))) 144 ∀x (¬M (x) ∨ ¬F (x)) 145 ¬∃x (M (x) ∧ ¬F (x)) 146 ¬∃x (M (x) ∧ F (x)) 147 ¬∀x (F (x) → ¬M (x)) In 148–151 let F (A) be the predicate “A is a finite set” and S(A, B) be the predicate “A is contained in B” Suppose the universe of discourse consists of all sets Translate the statement into symbols 148 Not all sets are finite 149 Every subset of a finite set is finite 150 No infinite set is contained in a finite set 151 The empty set is a subset of every finite set In 152–156 write the negation of the statement in good English Don’t write “It is not true that ” 152 Some bananas are yellow 153 All integers ending in the digit are odd 154 No tests are easy 155 Roses are red and violets are blue 156 Some skiers not speak Swedish 157 A student is asked to give the negation of “all bananas are ripe” (a) The student responds “all bananas are not ripe” Explain why the English in the student’s response is ambiguous (b) Another student says that the negation of the statement is “no bananas are ripe” Explain why this is not correct (c) Another student says that the negation of the statement is “some bananas are ripe” Explain why this is not correct (d) Give the correct negation 158 Explain why the negation of “Some students in my class use e-mail” is not “Some students in my class not use e-mail” 159 What is the rule of inference used in the following: If it snows today, the university will be closed The university will not be closed today Therefore, it did not snow today 472 Test Bank Questions and Answers 160 What is the rule of inference used in the following: If I work all night on this homework, then I can answer all the exercises If I answer all the exercises, I will understand the material Therefore, if I work all night on this homework, then I will understand the material 161 Explain why an argument of the following form is not valid: p→q ¬p ¬q 162 Determine whether the following argument is valid: p→r q→r ¬(p ∨ q) .¬r 163 Determine whether the following argument is valid: p→r q→r q ∨ ¬r .¬p 164 Show that the hypotheses “I left my notes in the library or I finished the rough draft of the paper” and “I did not leave my notes in the library or I revised the bibliography” imply that “I finished the rough draft of the paper or I revised the bibliography” 165 Determine whether the following argument is valid Name the rule of inference or the fallacy If n is a real number such that n > 1, then n2 > Suppose that n2 > Then n > 166 Determine whether the following argument is valid Name the rule of inference or the fallacy If n is a real number such that n > 2, then n2 > Suppose that n ≤ Then n2 ≤ 167 Determine whether the following argument is valid: She is a Math Major or a Computer Science Major If she does not know discrete math, she is not a Math Major If she knows discrete math, she is smart She is not a Computer Science Major Therefore, she is smart 168 Determine whether the following argument is valid Rainy days make gardens grow Gardens don’t grow if it is not hot It always rains on a day that is not hot Therefore, if it is not hot, then it is hot 169 Determine whether the following argument is valid If you are not in the tennis tournament, you will not meet Ed If you aren’t in the tennis tournament or if you aren’t in the play, you won’t meet Kelly You meet Kelly or you don’t meet Ed It is false that you are in the tennis tournament and in the play Therefore, you are in the tennis tournament 170 Show that the premises “Every student in this class passed the first exam” and “Alvina is a student in this class” imply the conclusion “Alvina passed the first exam” 171 Show that the premises “Jean is a student in my class” and “No student in my class is from England” imply the conclusion “Jean is not from England” 473 Test Bank Questions and Answers 172 Determine whether the premises “Some math majors left the campus for the weekend” and “All seniors left the campus for the weekend” imply the conclusion “Some seniors are math majors.” 173 Show that the premises “Everyone who read the textbook passed the exam”, and “Ed read the textbook” imply the conclusion “Ed passed the exam” 174 Determine whether the premises “No juniors left campus for the weekend” and “Some math majors are not juniors” imply the conclusion “Some math majors left campus for the weekend.” 175 Show that the premise “My daughter visited Europe last week” implies the conclusion “Someone visited Europe last week” 176 Suppose you wish to prove a theorem of the form “if p then q” (a) If you give a direct proof, what you assume and what you prove? (b) If you give a proof by contraposition, what you assume and what you prove? (c) If you give a proof by contradiction, what you assume and what you prove? 177 Suppose that you had to prove a theorem of the form “if p then q” Explain the difference between a direct proof and a proof by contraposition 178 Give a direct proof of the following: “If x is an odd integer and y is an even integer, then x + y is odd” 179 Give a proof by contradiction of the following: “If n is an odd integer, then n2 is odd” 180 Consider the following theorem: “if x and y are odd integers, then x + y is even” Give a direct proof of this theorem 181 Consider the following theorem: “if x and y are odd integers, then x+y is even” Give a proof by contradiction of this theorem 182 Give a proof by contradiction of the following: If x and y are even integers, then xy is even 183 Consider the following theorem: If x is an odd integer, then x + is odd Give a direct proof of this theorem 184 Consider the following theorem: If x is an odd integer, then x + is odd Give a proof by contraposition of this theorem 185 Consider the following theorem: If x is an odd integer, then x + is odd Give a proof by contradiction of this theorem 186 Consider the following theorem: If n is an even integer, then n + is odd Give a direct proof of this theorem 187 Consider the following theorem: If n is an even integer, then n + is odd Give a proof by contraposition of this theorem 188 Consider the following theorem: If n is an even integer, then n + is odd Give a proof by contradiction of this theorem 189 Prove that the following is true for all positive integers n: n is even if and only if 3n2 + is even 190 Prove the following theorem: n is even if and only if n2 is even 191 Prove: if m and n are even integers, then mn is a multiple of 192 Prove or disprove: For all real numbers x and y, x − y = x − y 193 Prove or disprove: For all real numbers x and y, x + x = 2x 194 Prove or disprove: For all real numbers x and y, xy = x · y 195 Give a proof by cases that x ≤ |x| for all real numbers x Test Bank Questions and Answers 481 Prove that A ∩ B = A ∪ B by giving a Venn diagram proof Prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side) 10 Prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) by giving an element table proof 11 Prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) by giving a proof using logical equivalence 12 Prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) by giving a Venn diagram proof 13 Prove or disprove: if A, B, and C are sets, then A − (B ∩ C) = (A − B) ∩ (A − C) 14 Prove or disprove A ⊕ (B ⊕ C) = (A ⊕ B) ⊕ C In questions 15–18 use a Venn diagram to determine which relationship, ⊆, =, or ⊇, is true for the pair of sets 15 A ∪ B, A ∪ (B − A) 16 A ∪ (B ∩ C), (A ∪ B) ∩ C 17 (A − B) ∪ (A − C), A − (B ∩ C) 18 (A − C) − (B − C), A − B In questions 19–23 determine whether the given set is the power set of some set If the set is a power set, give the set of which it is a power set 19 {Ø, {Ø}, {a}, {{a}}, {{{a}}}, {Ø, a}, {Ø, {a}}, {Ø, {{a}}}, {a, {a}}, {a, {{a}}}, {{a}, {{a}}}, {Ø, a, {a}}, {Ø, a, {{a}}}, {Ø, {a}, {{a}}}, {a, {a}, {{a}}}, {Ø, a, {a}, {{a}}}} 20 {Ø, {a}} 21 {Ø, {a}, {Ø, a}} 22 {Ø, {a}, {Ø}, {a, Ø}} 23 {Ø, {a, Ø}} 24 Prove that S ∪ T = S ∩ T for all sets S and T In 25–35 mark each statement TRUE or FALSE Assume that the statement applies to all sets 25 A − (B − C) = (A − B) − C 26 (A − C) − (B − C) = A − B 27 A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) 28 A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C) 29 A ∪ B ∪ A = A 30 If A ∪ C = B ∪ C, then A = B 31 If A ∩ C = B ∩ C, then A = B 32 If A ∩ B = A ∪ B, then A = B 33 If A ⊕ B = A, then B = A 34 There is a set A such that |P(A)| = 12 482 Test Bank Questions and Answers 35 A ⊕ A = A 36 Find three subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9} such that the intersection of any two has size and the intersection of all three has size 37 Find +∞ [−1/i, 1/i ] i=1 38 Find +∞ i=1 39 Find +∞ i=1 40 Find +∞ i=1 (1 − 1i , ) [1 − 1i , ] (i, ∞ ) 41 Suppose U = {1, 2, , 9}, A = all multiples of 2, B = all multiples of 3, and C = {3, 4, 5, 6, 7} Find C − (B − A) 42 Suppose S = {1, 2, 3, 4, 5} Find |P(S)| In questions 43–46 suppose A = {x, y} and B = {x, {x}} Mark the statement TRUE or FALSE 43 x ⊆ B 44 Ø ∈ P(B) 45 {x} ⊆ A − B 46 |P(A)| = In questions 47–54 suppose A = {a, b, c} Mark the statement TRUE or FALSE 47 {b, c} ∈ P(A) 48 {{a}} ⊆ P(A) 49 Ø ⊆ A 50 {Ø} ⊆ P(A) 51 Ø ⊆ A × A 52 {a, c} ∈ A 53 {a, b} ∈ A × A 54 (c, c) ∈ A × A In questions 55–62 suppose A = {1, 2, 3, 4, 5} Mark the statement TRUE or FALSE 55 {1} ∈ P(A) 56 {{3}} ⊆ P(A) 57 Ø ⊆ A 58 {Ø} ⊆ P(A) 59 Ø ⊆ P(A) 60 {2, 4} ∈ A × A Test Bank Questions and Answers 61 {Ø} ∈ P(A) 62 (1, 1) ∈ A × A In questions 63–65 suppose the following are fuzzy sets: F = {0.7 Ann, 0.1 Bill, 0.8 Fran, 0.3 Olive, 0.5 Tom}, R = {0.4 Ann, 0.9 Bill, 0.9 Fran, 0.6 Olive, 0.7 Tom} 63 Find F and R 64 Find F ∪ R 65 Find F ∩ R In questions 66–75, suppose A = {a, b, c} and B = {b, {c}} Mark the statement TRUE or FALSE 66 c ∈ A − B 67 |P(A × B)| = 64 68 Ø ∈ P(B) 69 B ⊆ A 70 {c} ⊆ B 71 {a, b} ∈ A × A 72 {b, c} ∈ P(A) 73 {b, {c}} ∈ P(B) 74 Ø ⊆ A × A 75 {{{c}}} ⊆ P(B) 76 Find A2 if A = {1, a} In questions 77–89 determine whether the set is finite or infinite If the set is finite, find its size 77 { x | x ∈ Z and x2 < 10 } 78 P({a, b, c, d}), where P denotes the power set 79 {1, 3, 5, 7, } 80 A × B, where A = {1, 2, 3, 4, 5} and B = {1, 2, 3} 81 { x | x ∈ N and 9x2 − = } 82 P(A), where A is the power set of {a, b, c} 83 A × B, where A = {a, b, c} and B = Ø 84 { x | x ∈ N and 4x2 − = } 85 { x | x ∈ Z and x2 = } 86 P(A), where A = P({1, 2}) 87 {1, 10, 100, 1000, } 88 S × T , where S = {a, b, c} and T = {1, 2, 3, 4, 5} 483 484 Test Bank Questions and Answers 89 { x | x ∈ Z and x2 < } 90 Prove that between every two rational numbers a/b and c/d (a) there is a rational number (b) there are an infinite number of rational numbers 91 Prove that there is no smallest positive rational number 92 Consider these functions from the set of licensed drivers in the state of New York Is a function one-to-one if it assigns to a licensed driver his or her (a) birthdate (b) mother’s first name (c) drivers license number? In 93–94 determine whether each of the following sets is countable or uncountable For those that are countably infinite exhibit a one-to-one correspondence between the set of positive integers and that set 93 The set of positive rational numbers that can be written with denominators less than √ 94 The set of irrational numbers between and π/2 95 Adapt the Cantor diagonalization argument to show that the set of positive real numbers less than with decimal representations consisting only of 0s and 1s is uncountable 96 Show that (0, 1) has the same cardinality as (0, 2) 97 Show that (0, 1] and R have the same cardinality In questions 98–106 determine whether the rule describes a function with the given domain and codomain √ 98 f : N → N where f (n) = n √ 99 h: R → R where h(x) = x 100 g: N → N where g(n) = any integer > n 101 F : R → R where F (x) = x−5 102 F : Z → R where F (x) = x2 103 F : Z → Z where F (x) = x2 104 G: R → R where G(x) = 105 f : R → R where f (x) = −5 −5 x + if x ≥ x − if x ≤ x2 if x ≤ x − if x ≥ 106 G: Q → Q where G(p/q) = q 107 Give an example of a function f : Z → Z that is 1-1 and not onto Z 108 Give an example of a function f : Z → Z that is onto Z but not 1-1 109 Give an example of a function f : Z → N that is both 1-1 and onto N 110 Give an example of a function f : N → Z that is both 1-1 and onto Z 111 Give an example of a function f : Z → N that is 1-1 and not onto N 485 Test Bank Questions and Answers 112 Give an example of a function f : N → Z that is onto Z and not 1-1 113 Suppose f : N → N has the rule f (n) = 4n + Determine whether f is 1-1 114 Suppose f : N → N has the rule f (n) = 4n + Determine whether f is onto N 115 Suppose f : Z → Z has the rule f (n) = 3n2 − Determine whether f is 1-1 116 Suppose f : Z → Z has the rule f (n) = 3n − Determine whether f is onto Z 117 Suppose f : N → N has the rule f (n) = 3n2 − Determine whether f is 1-1 118 Suppose f : N → N has the rule f (n) = 4n2 + Determine whether f is onto N 119 Suppose f : R → R where f (x) = x/2 (a) Draw the graph of f (b) Is f 1-1? 120 Suppose f : R → R where f (x) = x/2 (a) If S = { x | ≤ x ≤ }, find f (S) (c) Is f onto R? (b) If T = {3, 4, 5}, find f −1 (T ) 121 Determine whether f is a function from the set of all bit strings to the set of integers if f (S) is the position of a bit in the bit string S 122 Determine whether f is a function from the set of all bit strings to the set of integers if f (S) is the number of bits in S 123 Determine whether f is a function from the set of all bit strings to the set of integers if f (S) is the largest integer i such that the ith bit of S is and f (S) = when S is the empty string (the string with no bits) 124 Let f (x) = x3 /3 Find f (S) if S is: (a) {−2, −1, 0, 1, 2, 3} (b) {0, 1, 2, 3, 4, 5} 125 Suppose f : R → Z where f (x) = 2x − (a) Draw the graph of f (b) Is f 1-1? (Explain) 126 Suppose f : R → Z where f (x) = 2x − (a) If A = {x | ≤ x ≤ 4}, find f (A) (c) If C = {−9, −8}, find f −1 (C) 127 Suppose g: R → R where g(x) = (a) Draw the graph of g (c) {1, 5, 7, 11} x−1 (b) Is g 1-1? x−1 (a) If S = {x | ≤ x ≤ 6}, find g(S) (d) {2, 6, 10, 14} (c) Is f onto Z? (Explain) (b) If B = {3, 4, 5, 6, 7}, find f (B) (d) If D = {0.4, 0.5, 0.6}, find f −1 (D) (c) Is g onto R? 128 Suppose g: R → R where g(x) = (b) If T = {2}, find g −1 (T ) 129 Show that x = − −x 130 Prove or disprove: For all positive real numbers x and y, x · y ≤ x · y 131 Prove or disprove: For all positive real numbers x and y, x · y ≤ x · y 132 Suppose g: A → B and f : B → C where A = {1, 2, 3, 4}, B = {a, b, c}, C = {2, 7, 10}, and f and g are defined by g = {(1, b), (2, a), (3, a), (4, b)} and f = {(a, 10), (b, 7), (c, 2)} Find f ◦ g 133 Suppose g: A → B and f : B → C where A = {1, 2, 3, 4}, B = {a, b, c}, C = {2, 7, 10}, and f and g are defined by g = {(1, b), (2, a), (3, a), (4, b)} and f = {(a, 10), (b, 7), (c, 2)} Find f −1 486 Test Bank Questions and Answers In questions 134–137 suppose that g: A → B and f : B → C where A = B = C = {1, 2, 3, 4}, g = {(1, 4), (2, 1), (3, 1), (4, 2)}, and f = {(1, 3), (2, 2), (3, 4), (4, 2)} 134 Find f ◦ g 135 Find g ◦ f 136 Find g ◦ g 137 Find g ◦ (g ◦ g) In questions 138–141 suppose g: A → B and f : B → C where A = {1, 2, 3, 4}, B = {a, b, c}, C = {2, 8, 10}, and g and f are defined by g = {(1, b), (2, a), (3, b), (4, a)} and f = {(a, 8), (b, 10), (c, 2)} 138 Find f ◦ g 139 Find f −1 140 Find f ◦ f −1 141 Explain why g −1 is not a function In questions 142–143 suppose g: A → B and f : B → C where A = {a, b, c, d}, B = {1, 2, 3}, C = {2, 3, 6, 8}, and g and f are defined by g = {(a, 2), (b, 1), (c, 3), (d, 2)} and f = {(1, 8), (2, 3), (3, 2)} 142 Find f ◦ g 143 Find f −1 144 For any function f : A → B, define a new function g: P(A) → P(B) as follows: for every S ⊆ A, g(S) = { f (x) | x ∈ S } Prove that f is onto if and only if g is onto In questions 145–149 find the inverse of the function f or else explain why the function has no inverse 145 f : Z → Z where f (x) = x mod 10 146 f : A → B where A = {a, b, c}, B = {1, 2, 3} and f = {(a, 2), (b, 1), (c, 3)} 147 f : R → R where f (x) = 3x − 148 f : R → R where f (x) = 2x 149 f : Z → Z where f (x) = x − if x ≥ x + if x ≤ 150 Suppose g: A → B and f : B → C, where f ◦ g is 1-1 and g is 1-1 Must f be 1-1? 151 Suppose g: A → B and f : B → C, where f ◦ g is 1-1 and f is 1-1 Must g be 1-1? 152 Suppose f : R → R and g: R → R where g(x) = 2x + and g ◦ f (x) = 2x + 11 Find the rule for f In questions 153–157 for each partial function, determine its domain, codomain, domain of definition, set of values for which it is undefined or if it is a total function: 153 f : Z → R where f (n) = 1/n 154 f : Z → Z where f (n) = n/2 155 f : Z × Z → Q where f (m, n) = m/n 156 f : Z × Z → Z where f (m, n) = mn 157 f : Z × Z → Z where f (m, n) = m − n if m > n Test Bank Questions and Answers 487 , determine its domain, codomain, domain n − m2 of definition, and set of values for which it is undefined or whether it is a total function 158 For the partial function f : Z×Z → R defined by f (m, n) = 159 Let f : {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5, 6} be a function (a) How many total functions are there? (b) How many of these functions are one-to-one? In questions 160–166 find a formula that generates the following sequence a1 , a2 , a3 160 5, 9, 13, 17, 21, 161 3, 3, 3, 3, 3, 162 15, 20, 25, 30, 35, 163 1, 0.9, 0.8, 0.7, 0.6, 164 1, 1/3, 1/5, 1/7, 1/9, 165 2, 0, 2, 0, 2, 0, 2, 166 0, 2, 0, 2, 0, 2, 0, In questions 167–178, describe each sequence recursively Include initial conditions and assume that the sequences begin with a1 167 an = 5n 168 The Fibonacci numbers 169 0, 1, 0, 1, 0, 1, 170 an = + + + · · · + n 171 3, 2, 1, 0, −1, −2, 172 an = n! 173 1/2, 1/3, 1/4, 1/5, 174 0.1, 0.11, 0.111, 0.1111, 175 12 , 22 , 32 , 42 , 176 1, 111, 11111, 1111111, 177 an = the number of subsets of a set of size n 178 1, 101, 10101, 1010101, 179 Verify that an = is a solution to the recurrence relation an = 4an−1 − 3an−2 180 Verify that an = 3n is a solution to the recurrence relation an = 4an−1 − 3an−2 181 Verify that an = 3n+4 is a solution to the recurrence relation an = 4an−1 − 3an−2 182 Verify that an = 3n + is a solution to the recurrence relation an = 4an−1 − 3an−2 183 Verify that an = · 3n − π is a solution to the recurrence relation an = 4an−1 − 3an−2 In questions 184–188 find a recurrence relation with initial condition(s) satisfied by the sequence Assume a0 is the first term of the sequence 488 Test Bank Questions and Answers 184 an = 2n 185 an = 2n + 186 an = (−1)n 187 an = 3n − √ 188 an = 189 You take a job that pays $25,000 annually (a) How much you earn n years from now if you receive a three percent raise each year? (b) How much you earn n years from now if you receive a five percent raise each year? (c) How much you earn n years from now if each year you receive a raise of $1000 plus two percent of your previous year’s salary 190 Suppose inflation continues at three percent annually (That is, an item that costs $1.00 now will cost $1.03 next year.) Let an = the value (that is, the purchasing power) of one dollar after n years (a) Find a recurrence relation for an (b) What is the value of $1.00 after 20 years? (c) What is the value of $1.00 after 80 years? (d) If inflation were to continue at ten percent annually, find the value of $1.00 after 20 years (e) If inflation were to continue at ten percent annually, find the value of $1.00 after 80 years 191 Find the sum 1/4 + 1/8 + 1/16 + 1/32 + · · · 192 Find the sum + + + 16 + 32 + · · · + 228 193 Find the sum − + − 16 + 32 − · · · − 228 194 Find the sum − 1/2 + 1/4 − 1/8 + 1/16 − · · · 195 Find the sum + 1/2 + 1/8 + 1/32 + 1/128 + · · · 196 Find the sum 112 + 113 + 114 + · · · + 673 197 Find i=1 ((−2)i − 2i ) j 198 Find ij j=1 i=1 199 Rewrite (i2 + 1) so that the index of summation has lower limit and upper limit i=−3 200 Find a × matrix A= 0 0 such that A2 = 0 201 Suppose A is a × matrix, B is an × matrix, and C is a × matrix Find the number of rows, the number of columns, and the number of entries in A(BC) 202 Let A = m Find An where n is a positive integer 203 Suppose A = exists and C = Find a matrix B such that AB = C or prove that no such matrix 489 Test Bank Questions and Answers 204 Suppose B = exists 205 Suppose B = exists and C = Find a matrix A such that AB = C or prove that no such matrix 6 and C = Find a matrix A such that AB = C or prove that no such matrix In questions 206–212 determine whether the statement is true or false 206 If AB = AC, then B = C 207 If A = , then A−1 = 208 If A = , then A2 = −5 2 −3 25 209 If A is a × matrix and B is a × matrix, then AB has 16 entries 210 If A and B are × matrices such that AB= 0 0 , then A= 0 0 or B= 0 211 If A and B are × matrices, then A+B=B+A 212 AB=BA for all × matrices A and B  213 Suppose A =    1  and B =  1 (a) the join of A and B 1   Find (b) the meet of A and B (c) the Boolean product of A and B 214 Suppose A is a × matrix with real number entries such that AB=BA for all × matrices What relationships must exist among the entries of A? Answers for Chapter The first is a subset of the second, but the second is not a subset of the first The second is a subset of the first, but the first is not a subset of the second Neither is a subset of the other True, since A − (B ∩ C) = A ∩ B ∩ C = A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) = (A − B) ∪ (A − C) A ∩ B ⊆ A ∪ B: Let x ∈ A ∩ B x ∈ / A ∩ B, x ∈ / A or x ∈ / B, x ∈ A or x ∈ B, x ∈ A ∪ B Reversing the steps shows that A ∪ B ⊆ A ∩ B The columns for A ∩ B and A ∪ B match: each entry is if and only if A and B have the value A ∩ B = {x | x ∈ A ∩ B} = {x | x ∈ / A ∩ B} = {x | ¬(x ∈ A ∩ B)} = {x | ¬(x ∈ A ∧ x ∈ B)} = {x | ¬(x ∈ A) ∨ ¬(x ∈ B)} = {x | x ∈ /A ∨ x∈ / B} = {x | x ∈ A ∨ x ∈ B} = {x | x ∈ A ∪ B} = A ∪ B 490 Test Bank Questions and Answers A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C): Let x ∈ A ∩ (B ∪ C) .x ∈ A and x ∈ B ∪ C, x ∈ A and x ∈ B, or x ∈ A and x ∈ C, x ∈ (A ∩ B) ∪ (A ∩ C) Reversing the steps gives the opposite containment 10 Each set has the same values in the element table: the value is if and only if A has the value and either B or C has the value 11 A∩(B ∪C) = {x | x ∈ A∩(B ∪C)} = {x | x ∈ A ∧ x ∈ (B ∪C)} = {x | x ∈ A ∧ (x ∈ B ∨ x ∈ C)} = {x | (x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C)} = {x | x ∈ A∩B ∨ x ∈ A∩C} = {x | x ∈ (A∩B) ∪ (A∩C)} = (A∩B) ∪ (A∩C) 12 13 False For example, let A = {1, 2}, B = {1}, C = {2} 14 True, using either a membership table or a containment proof, for example 15 = 16 ⊇ 17 = 18 ⊆ 19 Yes {Ø, a, {a}, {{a}}} 20 Yes, {a} 21 No, it lacks {Ø} 22 Yes, {{a, Ø}} 23 No, it lacks {a} and {Ø} 24 Since S ∪ T = S ∩ T (De Morgan’s law), the complements are equal 25 False 26 False 27 True 28 False 29 True 30 False Test Bank Questions and Answers 31 False 32 True 33 False 34 False 35 False 36 For example, {1, 2, 3}, {2, 3, 4}, {1, 3, 4} 37 [−1, 1] 38 Ø 39 {1} 40 Ø 41 {4, 5, 6, 7} 42 32 43 False 44 True 45 False 46 True 47 True 48 True 49 True 50 True 51 True 52 True 53 False 54 True 55 True 56 True 57 True 58 True 59 True 60 False 61 False 62 True 63 F = {0.3 Ann, 0.9 Bill, 0.2 Fran, 0.7 Olive, 0.5 Tom}, R = {0.6 Ann, 0.1 Bill, 0.1 Fran, 0.4 Olive, 0.3 Tom} 64 {0.7 Ann, 0.9 Bill, 0.9 Fran, 0.6 Olive, 0.7 Tom} 65 {0.4 Ann, 0.1 Bill, 0.8 Fran, 0.3 Olive, 0.5 Tom} 66 True 67 True 68 True 69 False 70 False 71 False 72 True 491 492 Test Bank Questions and Answers 73 True 74 True 75 True 76 A2 = {(1, 1), (1, a), (a, 1), (a, a)} 77 78 16 79 Infinite 80 15 81 82 256 83 84 85 86 16 87 Infinite 88 15 89 90 (a) Assume (b) Assume 91 If < a b, a b a b < dc Then a b < a c b+d = ad+bc 2bd < dc < dc Let m1 be the midpoint of then < < a 4b < a 3b < a 2b < a c b, d For i > let mi be the midpoint of a b , mi−1 a b 92 (a) No (b) No (c) Yes 93 Countable To find a correspondence, follow the path in Example in Section 2.5, using only the first three lines 94 Uncountable 95 Assume that these numbers are countable, and list them in order r1 , r2 , r3 , Then form a new number r, whose i-th decimal digit is 0, if the i-th decimal digit of ri is 1, and whose i-th decimal digit is 1, if the i-th decimal digit of ri is Clearly r is not in the list r1 , r2 , r3 , , therefore the original assumption is false 96 The function f (x) = 2x is one-to-one and onto 97 Example 2.5.6 shows that |(0, 1]| = |(0, 1)|, and Exercise 2.5.34 shows that |(0, 1]| =R 98 Not a function; f (2) is not an integer 99 Function 100 Not a function; g(1) has more than one value 101 Not a function; F (5) not defined 102 Function 103 Not a function; F (1) not an integer 104 Not a function; the cases overlap For example, G(1) is equal to both and 105 Not a function; f (3) not defined 106 Not a function; f (1/2) = and f (2/4) = 107 f (n) = 2n 108 f (n) = n/2 109 f (n) = −2n, n≤0 2n − 1, n > 110 111 112 113 Test Bank Questions and Answers  −n   , n even f (n) = n2+   , n odd −2n, n≤0 f (n) = 2n + 1, n >  −n   , n even f (n) = n2−   , n odd Yes 114 No 115 No 116 No 117 Yes 118 No 119 (a) (b) No (c) No 120 (a) {0, 1, 2, 3} (b) [6, 12) 121 No; there may be no bits or more than one bit 122 Yes 123 No; f not defined for the string of all 1’s, for example S = 11111 124 (a) {−3, −1, 0, 2, 9} (b) {0, 2, 9, 21, 41} (c) {0, 41, 114, 443} (d) {2, 72, 333, 914} 125 (a) (b) No (c) Yes 126 (a) {1, 2, 3, 4, 5, 6, 7} (b) {5, 7, 9, 11, 13} (c) (−9/2, −7/2] (d) Ø 127 (a) 493 494 Test Bank Questions and Answers (b) No (c) No 128 (a) {0, 1, 2} (b) [5, 7) 129 Let n = x , so that n − < x ≤ n Multiplying by −1 yields −n + > −x ≥ −n, which means that −n = −x 130 False: x = y = 1.5 131 True: x ≤ x , y ≤ y ; therefore xy ≤ x y ; since x y is an integer at least as great as xy, then xy ≤ x y 132 {(1, 7), (2, 10), (3, 10), (4, 7)} 133 {(2, c), (7, b), (10, a)} 134 {(1, 2), (2, 3), (3, 3), (4, 2)} 135 {(1, 1), (2, 1), (3, 2), (4, 1)} 136 {(1, 2), (2, 4), (3, 4), (4, 1)} 137 {(1, 1), (2, 2), (3, 2), (4, 4)} 138 {(1, 10), (2, 8), (3, 10), (4, 8)} 139 {(2, c), (8, a), (10, b)} 140 {(2, 2), (8, 8), (10, 10)} 141 g −1 (a) is equal to both and 142 {(a, 3), (b, 8), (c, 2), (d, 3)} 143 {(2, 3), (3, 2), (8, 1)} 144 Suppose f is onto Let T ∈ P(B) and let S = { x ∈ A | f (x) ∈ T } Then g(S) = T , and g is onto If f is not onto B, let y ∈ B − f (A) Then there is no subset S of A such that g(S) = {y} 145 f −1 (10) does not exist 146 {(1, b), (2, a), (3, c)} 5+x 147 f −1 (x) = 148 f −1 ( 12 ) does not exist 149 f −1 (5) is not a single value 150 No 151 Yes 152 f (x) = x + 153 Z, R, Z − {0}, {0} 154 Z, Z, Z, total function 155 Z × Z, Q, Z × (Z − {0}), Z × {0} 156 Z × Z, Z, Z × Z, total function 157 Z × Z, Z, { (m, n) | m > n }, { (m, n) | m ≤ n } 158 Z × Z, R, { (m, n) | m = n or m = −n }, { (m, n) | m = n or m = −n } 159 (a) 65 = 7,776 (b) · · · · = 720 160 an = 4n + 161 an = 162 an = 5(n + 2) 163 an = − (n − 1)/10 495 Test Bank Questions and Answers 164 an = 1/(2n − 1) 165 an = + (−1)n+1 166 an = + (−1)n 167 an = 5an−1 , a1 = 168 an = an−1 + an−2 , a1 = a2 = 169 an = an−2 , a1 = 0, a2 = 170 an = an−1 + n, a1 = 171 an = an−1 − 1, a1 = 172 an = nan−1 , a1 = an−1 173 an = , a1 = 1/2 + an−1 174 an = an−1 + 1/10n , a1 = 0.1 175 an = an−1 + 2n − 1, a1 = 176 an = 100an−1 + 11 177 an = · an−1 , a1 = 178 an = 100an−1 + 1, a1 = 179 · − · = · = 180 · 3n−1 − · 3n−2 = · 3n−1 − 3n−1 = · 3n−1 = 3n 181 · 3n+3 − · 3n+2 = · 3n+3 − 3n+3 = · 3n+3 = 3n+4 182 4(3n−1 + 1) − 3(3n−2 + 1) = · 3n−1 − 3n−1 + − = 3n−1 (4 − 1) + = 3n + 183 4(7 · 3n−1 − π) − 3(7 · 3n−2 − π) = 28 · 3n−1 − · 3n−1 − 4π + 3π = · 3n − π 184 an = 2an−1 , a0 = 185 an = 2an−1 − 1, a0 = 186 an = −an−1 , a0 = 187 an = an−1 + 3, a0 = −1 √ 188 an = an−1 , a0 = 189 (a) 25, 000 · 1.03n 190 (a) an = an−1 /1.03 (e) 1/1.180 ≈ 0.00 (b) 25, 000 · 1.05n (b) a20 = 1/1.03 20 (c) 25, 000 · 1.02n + 1, 000 ≈ 0.55 191 1/2 192 229 − 193 + 23 (229 ) 194 2/3 195 8/3 196 220,585 197 −84 198 25 199 i=0 ((i − 3)2 + 1) 200 A matrix of the form −2a a −4a 2a where a = 201 A(BC) has rows, columns, and 54 entries 202 An = mn 1.02n −1 0.02 80 (c) a80 = 1/1.03 ≈ 0.09 (d) 1/1.120 ≈ 0.15 ... digit are not odd 154 Some tests are easy 155 Roses are not red or violets are not blue 156 All skiers speak Swedish Test Bank Questions and Answers Test Bank Questions and Answers 479 157 (a) Depending... 10}, and f and g are defined by g = {(1, b), (2, a), (3, a), (4, b)} and f = {(a, 10), (b, 7), (c, 2)} Find f −1 486 Test Bank Questions and Answers In questions 134–137 suppose that g: A → B and. .. be false and p and r be true 16 Not equivalent Let p, q, and r be false 17 Both truth tables are identical: p T T F F q T F T F (p → q) ∧ (¬p → q) T F T F q T F T F Test Bank Questions and Answers